Arthur L.B. Yang ; Philip B. Zhang
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The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler Theorem
dmtcs:2510 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2015,
DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
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https://doi.org/10.46298/dmtcs.2510
The Real-rootedness of Eulerian Polynomials via the Hermite–Biehler TheoremArticle
Authors: Arthur L.B. Yang 1; Philip B. Zhang 1
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Arthur L.B. Yang;Philip B. Zhang
1 Center for Combinatorics [Nankai]
Based on the Hermite–Biehler theorem, we simultaneously prove the real-rootedness of Eulerian polynomials of type $D$ and the real-rootedness of affine Eulerian polynomials of type $B$, which were first obtained by Savage and Visontai by using the theory of $s$-Eulerian polynomials. We also confirm Hyatt’s conjectures on the inter-lacing property of half Eulerian polynomials. Borcea and Brändén’s work on the characterization of linear operators preserving Hurwitz stability is critical to this approach.